Page:Elementary Principles in Statistical Mechanics (1902).djvu/218

194 respect to exchange of energy and exchange of particles, with other grand ensembles canonically distributed and having the same values of $$\Theta$$ and of the coefficients $$\mu_1$$, $$\mu_2$$, etc., when the circumstances are such that exchange of energy and of particles are possible, and when equilibrium would not subsist, were it not for equal values of these constants in the two ensembles.

With respect to the exchange of energy, the case is exactly the same as that of the petit ensembles considered in Chapter IV, and needs no especial discussion. The question of exchange of particles is to a certain extent analogous, and may be treated in a somewhat similar manner. Let us suppose that we have two grand ensembles canonically distributed with respect to specific phases, with the same value of the modulus and of the coefficients $$\mu_1\ldots \mu_h$$, and let us consider the ensemble of all the systems obtained by combining each system of the first ensemble with each of the second.

The probability-coefficient of a generic phase in the first ensemble may be expressed by The probability-coefficient of a specific phase will then be expressed by  since each generic phase comprises $${\nu_1}! \ldots {\nu_h}!$$ specific phases. In the second ensemble the probability-coefficients of the generic and specific phases will be and